We consider a Dirichlet double phase problem with unbalanced growth. In the reaction we have the combined effects of a critical term and of a locally defined Carathéodory perturbation. Using cut-off functions and truncation techniques we bypass the critical term and deal with a coercive problem. Using this auxillary problem, we show that the original Dirichlet equation has a whole sequence of nodal (sign-changing) solutions which converge to zero in the Musielak–Orlice–Sobolev space and in .
L.Gasiński and N.S.Papageorgiou, Nodal solutions for nonlinear nonhomogeneous Robin problems with an indefinite potential, Proc. Edinburgh Math. Soc.61 (2018), 943–959. doi:10.1017/S0013091518000044.
7.
L.Gasiński and N.S.Papageorgiou, On a nonlinear parametric Robin problem with a locally defined redction, Nonlinear Anal.185 (2019), 374–385. doi:10.1016/j.na.2019.03.019.
8.
L.Gasiński and P.Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal.195 (2020), 111739. doi:10.1016/j.na.2019.111739.
9.
B.Ge, D.J.Lv and J.F.Lu, Multiple solutions for a class of double phase problems without the Ambrosetti–Rabinowitz condition, Nonlinear Anal.188 (2019), 294–315. doi:10.1016/j.na.2019.06.007.
S.Hu and N.S.Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
12.
R.Kajikiya, A critical point theorem related to the symmetric mountain pass theorem and its applications to elliptic equations, J. Funct. Anal.225 (2005), 352–370. doi:10.1016/j.jfa.2005.04.005.
13.
S.Leonardi and N.S.Papageorgiou, On a class of critical Robin problems, Forum Math.32 (2020), 95–109.
14.
Z.Li and Z.Q.Wang, Schrödinger equations with concave and convex nonlinearities, ZAMP56 (2005), 609–629.
15.
W.Liu and G.Dai, Existence and multiplicity results for double phase problems, J. Differential Equ.265 (2018), 4311–4334. doi:10.1016/j.jde.2018.06.006.
16.
P.Marcellini, Reguarity and existence of solutions of elliptic equations with -growth coditions, J. Differential Equ.90 (1991), 1–30. doi:10.1016/0022-0396(91)90158-6.
17.
J.Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., Vol. 1034, Springer-Verlag, 1983.
18.
N.S.Papageorgiou and V.D.Rădulescu, Existence and multiplicity of solutions for double-phase Robin problems, Bull. London Math. Soc.52 (2020), 546–560. doi:10.1112/blms.12347.
19.
N.S.Papageorgiou, V.D.Rădulescu and D.Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discr. Cont. Dyn. Syst. A37 (2017), 2589–2618. doi:10.3934/dcds.2017111.
20.
N.S.Papageorgiou, V.D.Rădulescu and D.Repovš, Double phase problems with reaction of arbitrary growth, ZAMP69 (2018), 108.
21.
N.S.Papageorgiou, V.D.Rădulescu and D.Repovš, Nonlinear Analysis-Theory and Methods, Springer Nature, Swizerland AG, 2019.
22.
N.S.Papageorgiou, V.D.Rădulescu and D.Repovš, Nonlinear nonhomogeneous Robin problems with a indefinite potential and general reaction, Appl Math. Optim.81 (2020), 823–857. doi:10.1007/s00245-018-9521-x.
23.
N.S.Papageorgiou and A.Scapellato, Constant sign and nodal solutions for parametric -equations, Adv. Nonlinear Anal.9(1) (2020), 449–478. doi:10.1515/anona-2020-0009.
24.
N.S.Papageorgiou, C.Vetro and F.Vetro, Multiple solutions for parametric double phase Dirichlet problems, Comm. Contemp. Math. doi:10.1142/so219199720500066.
25.
N.S.Papageorgiou, C.Vetro and F.Vetro, Solutions for parametric double phase Robin problems, Asymptotic Anal. doi:10.3233/ASY-201598.
26.
N.S.Papageorgiou and P.Winkert, Applied Nonlinear Functional Analysis, DeGruyter, Berlin, 2018.
27.
N.S.Papageorgiou and C.Zhang, Noncoercive resonant -equations with concave terms, Adv. Nonlinear Anal.9(1) (2020), 228–249. doi:10.1515/anona-2018-0175.
28.
A.M.Ragusa and A.Tachikawa, Regularity for minimizers of double phase with variable exponents, Adv. Nonlin. Anal.9 (2020), 710–728. doi:10.1515/anona-2020-0022.
29.
Z.Q.Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlin. Diff. Equ. Appl. (NoDEA)8 (2001), 815–833.
30.
V.V.Zhikov, Averaging of functionals of the calculus of variations and elasticity, Math. USSR-Ieves.29 (1987), 33–66. doi:10.1070/IM1987v029n01ABEH000958.
31.
V.V.Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci.173 (2011), 463–570. doi:10.1007/s10958-011-0260-7.
